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A relaxation result for state constrained inclusions ininfinite dimension

Hélène Frankowska, Elsa Marchini, Marco Mazzola

To cite this version:Hélène Frankowska, Elsa Marchini, Marco Mazzola. A relaxation result for state constrained inclusionsin infinite dimension. Mathematical Control and Related Fields, AIMS, 2015, 6 (1), pp.113-141.<10.3934/mcrf.2016.6.113>. <hal-01099223>

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A RELAXATION RESULT FOR STATE CONSTRAINED INCLUSIONSIN INFINITE DIMENSION

H. FRANKOWSKA, E.M. MARCHINI, AND M. MAZZOLA

Abstract. In this paper we consider a state constrained differential inclusion x ∈ Ax+F (t, x), with A generator of a strongly continuous semigroup in an infinite dimensionalseparable Banach space. Under an “inward pointing condition” we prove a relaxationresult stating that the set of trajectories lying in the interior of the constraint is dense inthe set of constrained trajectories of the convexified inclusion x ∈ Ax + coF (t, x). Someapplications to control problems involving PDEs are given.

1. Introduction

We study a class of infinite dimensional differential inclusions subject to state constraints.Interest in these kind of equations arises in several contexts. Differential inclusions finda natural application in a research area of great development, the control theory, andthe infinite dimensional setting allows to apply our results to control problems involvingPDEs. Hence, models describing many physical phenomena such as diffusion, vibrationof strings, fluid dynamics, may be included in our analysis.

In this paper we are concerned with the differential inclusion

(1.1) x(t) ∈ Ax(t) + F (t, x(t)) a.e. t ∈ [t0, 1] ,

and the convexified differential inclusion

(1.2) x(t) ∈ Ax(t) + coF (t, x(t)) a.e. t ∈ [t0, 1] ,

with coF (t, x(t)) the closed convex hull of F (t, x(t)). The operator A is the infinitesimalgenerator of a strongly continuous semigroup S(t) : X → X, X is an infinite dimensionalseparable Banach space, F : I × X X is a set-valued map with closed non-emptyimages, I = [0, 1] and t0 ∈ I. The trajectories of the differential inclusion (1.1) areunderstood in the mild sense (see [25]) and are subject to the state constraint. Namelygiven a set K ⊂ X, we restrict our attention to the trajectories satisfying

(1.3) x(t) ∈ K, for t ∈ [t0, 1] .

In this paper we shall always assume that K is the closure of an open subset of X. Whensatisfying the constraint, a trajectory x is called feasible.

Differential inclusions, and control systems, in presence of state constraints, are largelyemployed in applied sciences. One of the tools playing a key role in this context consists inapproximating feasible trajectories by trajectories lying in the interior of the constraints.It is used for instance to establish regularity properties of value functions, to justify the

Date: January 1, 2015.This work was co-funded by the European Union under the 7th Framework Programme “FP7-

PEOPLE-2010-ITN”, grant agreement number 264735-SADCO.1

2 H. FRANKOWSKA, E.M. MARCHINI, AND M. MAZZOLA

use of the Maximum Principle in normal form, to prove existence and regularity resultsof optimal solutions. The classical technique employed to construct the approximatingtrajectories relies on the possibility of directing the velocity into the interior of the con-straint K whenever approaching the boundary of K. To this aim, in the finite dimensionalsetting, an “inward pointing condition” was proposed by Soner, see [28], to get continuityof the value function associated to an optimal control problem with dynamics x ∈ F (x)independent of t. Since then, this subject has received considerable attention, a partiallist of references includes [5, 6, 10, 16, 18, 19].

Defining the oriented distance from x ∈ X to K by

dK(x) =

infk∈K ‖x− k‖X if x /∈ K− infk∈(X\K) ‖x− k‖X otherwise,

the inward pointing condition, in the case of time independent F and state constraintswith a locally C1,1 boundary, takes the following form:

(1.4) minv∈F (x)

〈∇dK(x), v〉 ≤ −ρ, ∀ x ∈ ∂K ,

cf. [5, 18]. As in many applied models state constraints having nonsmooth boundary arepresent, a number of papers made extensions of (1.4) to the nonsmooth setting. However,contrary to the smooth case, here some regularity of the dynamics F (t, x) is usuallyrequired both in t and in x. On the other hand, it may happen in some applications thatthe dynamics depends on t in a merely measurable way. In order to extend the theory tothis situation, in the recent works [16, 17] a new inward pointing condition (equivalentto the classical one if K has smooth boundary) is proposed: for any “bad” velocity vpointing outside the constraint, there exists a “good” one v such that the difference v− vpoints inside in a uniform way, namely:

(1.5)

∀ x ∈ ∂K, ∃ ρ > 0, ∀ t ∈ I, ∀ v ∈ F (t, x) with sup

ξ∈∂dK(x)

〈ξ, v〉 ≥ 0,

∃ v ∈ F (t, x) satisfying supξ∈∂dK(x)

⟨ξ, v − v

⟩< −ρ .

Here ∂dK(x) denotes the Clarke generalized gradient of dK at the point x ∈ X. Under thisassumption, in [16, 17] some approximation results were proved in order to get uniquenessof solutions for a constrained Hamilton-Jacobi-Bellman equation.

The purpose of the present paper is to perform the analysis in the infinite dimensionalsetting, the natural framework for many phenomena described by PDEs. Also in this casewe need results which permit to approximate feasible trajectories by trajectories stayingin the interior of the state constraints. Assuming an inward pointing condition, Theorem3.2 below guarantees the existence of the required approximation. Notice that, althoughthe literature dealing with infinite dimensional control theory (and infinite dimensionaldifferential inclusions) is quite rich, see e.g. the books [2, 3, 4, 14, 21, 22], the recent paper[11] and the bibliography therein, to our knowledge, no similar results are known in thissetting. As an application, we obtain our main result, a relaxation theorem in infinitedimension (see Theorem 3.1).

We deal with great generality, allowing the state space X to be a separable Banachspace. Hence, our analysis applies to some interesting and delicate frameworks as the spaceof essentially bounded functions and the space of continuous functions. For this reason,

RELAXATION IN INFINITE DIMENSION 3

in this context, the relaxation theorem is obtained under a version of condition (1.5),requiring some uniformity on a neighborhood of ∂K and with respect to the semigroup.Nevertheless, as illustrated in Section 3, if some compactness conditions are satisfied, amuch more simple condition, analogue to the finite dimensional (1.5) is sufficient.

We consider the following inward pointing condition:

∀ x ∈ ∂K, ∃ η, ρ, M > 0 such that ∀ t ∈ I, ∀ x ∈ K ∩B(x, η),(1.6)

∀ v ∈ coF (t, x) satisfying supτ≤η, ξ∈∂dK(z0)

〈ξ, S(τ) v〉 ≥ 0, for some z0 ∈ B(x, η),

∃ v ∈ coF (t, x) ∩B(v,M) satisfying supz ∈ B(S(τ)x, η)τ ≤ η, ξ ∈ ∂dK(z)

⟨ξ, S(τ) (v − v)

⟩< −ρ.

Notice that condition (1.6) deals with the set-valued map coF , since, in order to provethe relaxation theorem, we must be able to approximate relaxed trajectories by relaxedtrajectories lying in the interior of K. However, under additional compactness conditions,the first convex hull can be removed from (1.6).

Quite interesting for the applications is the case when X is a Hilbert space, see Section5. In this framework we will provide an alternative version of condition (1.6), whichdrastically simplifies the analysis when the set of constraints K is convex. The inwardpointing condition needed here involves projections on convex sets rather than generalizedgradients of the oriented distance function which belong to the dual space X∗.

1.1. Outline of the paper. Section 2 contains a list of notations, definitions, and as-sumptions in use. The main theorems are stated in Section 3. Some results which allowto simplify the inward pointing condition are also proposed. The Hilbert setting is ana-lyzed in Section 4, while Section 5 is devoted to some applications involving PDEs andintegrodifferential equations.

2. Preliminaries

In this Section we list the notation and the main assumptions in use throughout the paper.

2.1. Notation.

- B(x, r) denotes the closed ball of center x ∈ X and radius r > 0; B is the closedunit ball in X centered at 0;

- given a Banach space Y , L(X, Y ) denotes the Banach space of bounded linearoperators from X into Y , C(I,X) the space of continuous functions from I to X,L1(I,X) the space of Bochner integrable functions from I to X, and L∞(I,X) thespace of measurable essentially bounded functions from I to X;

- 〈·, ·〉 stands for the duality pairing on X∗ ×X;- µ is the Lebesgue measure on the real line.

We will use the following notion of solution.

Definition 2.1. Let t0 ∈ I and x0 ∈ X. A function x ∈ C([t0, 1], X) is a (mild) solutionof (1.1) with initial datum x(t0) = x0 if there exists a function f ∈ L1([t0, 1], X) such that

(2.1) f(t) ∈ F (t, x(t)), for a.e. t ∈ (t0, 1)


and

(2.2) x(t) = S(t− t0)x0 +

∫ t

t0

S(t− s) f(s) ds, for any t ∈ [t0, 1],

i.e. f is an integrable selection of the set valued map t F (t, x(t)) and x is a mildsolution (see [25]) of the initial value problem

(2.3)

x(t) = Ax(t) + f(t), for a.e. t ∈ [t0, 1]x(t0) = x0.

In order to simplify the notation, for t0 ∈ I and x0 ∈ X, we denote by SK(t0, x0) the setof mild solutions x of (1.1) satisfying x(t0) = x0 and (1.3), and by fx the correspondingmeasurable selection in (2.3).

Notice that, since S(t) is a strongly continuous semigroup, there exists MS > 0 such that

(2.4) ‖S(t)‖L(X,X) ≤MS, for any t ∈ I.The differential inclusion (1.1) is a convenient tool to investigate for example the semi-

linear control system

(2.5)

x(t) = Ax(t) + f(t, x(t), u(t)) a.e. t ∈ [t0, 1]u(t) ∈ U

where U is an appropriate separable metric space of controls. Setting F (t, x) = f(t, x, U),we can reduce (2.5) to (1.1) by applying a measurable selection theorem.

2.2. Assumptions. In our main theorems, we will assume the following conditions:

- positive invariance of K by the semigroup:

(2.6) S(t)K ⊂ K ∀ t ∈ I;

- ∀ t ∈ I and any x ∈ X, F (t, x) is closed, and, for any x ∈ X,

(2.7) the set-valued map F (·, x) is Lebesgue measurable;

- F (t, ·) is locally Lipschitz in the following sense: for any R > 0, there existskR ∈ L1(I;R+) such that, for a.e. t ∈ I and any x, y ∈ RB,

(2.8) F (t, x) ⊂ F (t, y) + kR(t)‖x− y‖XB;

- there exists φ ∈ L1(I;R+) such that, for a.e. t ∈ I and any x ∈ X,

(2.9) F (t, x) ⊂ φ(t)(1 + ‖x‖X)B.

3. The main results

In this section we state the main results of the paper. The first is a relaxation theorem.

Theorem 3.1. Assume (1.6) and (2.6)–(2.9). Then, for any ε > 0 and any feasibletrajectory x of (1.2), (1.3), there exists a trajectory x of (1.1) satisfying

(3.1) x(t0) = x(t0), x(t) ∈ IntK, for any t ∈ (t0, 1]


and

(3.2) ‖x− x‖L∞(t0,1;X) ≤ ε.

The key point in the proof of Theorem 3.1, is a result on approximation of feasibletrajectories, by trajectories lying in the interior of the constraint K.

Theorem 3.2. Assume (2.6)–(2.9) and that


∀ v ∈ F (t, x) satisfying supτ≤η, ξ∈∂dK(z0)

〈ξ, S(τ) v〉 ≥ 0, for some z0 ∈ B(x, η),

∃ v ∈ F (t, x) ∩B(v,M) satisfying supz ∈ B(S(τ)x, η)τ ≤ η, ξ ∈ ∂dK(z)

⟨ξ, S(τ) (v − v)

⟩< −ρ.

Then, for any ε > 0 and any feasible trajectory x of (1.1), (1.3), there exists a trajectoryx of (1.1) satisfying (3.1) and (3.2).

In the following propositions pointwise versions of the inward pointing condition (3.3)are proposed, see the applications in Section 5.

Proposition 3.3. Assume that (2.8)–(2.9) hold true with time independent kR, φ ∈ R+,

F (·, x) is continuous for any x ∈ X,(3.4)

and

F (t, x) is compact, for any t ∈ I and any x ∈ ∂K.(3.5)

Then, assumption (1.5) implies (3.3). Consequently, if (1.5) holds true with F replacedby co F , then (1.6) is satisfied.

In the next proposition the convexity of values of F is needed on the boundary of K.

Proposition 3.4. Assume that (3.4) and (2.8)–(2.9), with time independent kR, φ ∈ R+,hold true, that X is reflexive, for any x ∈ ∂K and t ∈ I, F (t, x) is convex, and

the set-valued map ∂dK(·) is upper semicontinuous in x, and ∂dK(x) is compact.(3.6)

Then, assumption (1.5) implies (3.3) and (1.6).

Notice that when dK is C1 on a neighborhood of ∂K, then condition (3.6) is satisfied.In the proof of Theorem 3.1, we need to approximate relaxed trajectories by relaxedtrajectories lying in the interior of K. This is the reason why the inward pointing condition(1.6) required in this case involves the set-valued map coF . By the way, the first convexhull in (1.6) can be removed in some special cases, as indicated in the next proposition.

Proposition 3.5. Suppose that for every t ∈ I and x ∈ ∂K, coF (t, x) is closed and theset-valued map

(3.7) [0, 1] 3 t cov ∈ F (t, x) : sup

ξ∈∂dK(x)

〈ξ, v〉 ≤ 0

is upper semicontinuous with closed values. Assume that (3.4) and (2.8)–(2.9), with timeindependent kR, φ ∈ R+, hold true. Further, suppose that either (3.5) is satisfied, or that


X is a reflexive space and (3.6) is satisfied. Then, the following assumption: for anyx ∈ ∂K, there exists ρ > 0 such that

∀ t ∈ I and any v ∈ F (t, x) satisfying supξ∈∂dK(x)

〈ξ, v〉 ≥ 0,(3.8)

∃ v ∈ coF (t, x) satisfying supξ∈∂dK(x)

⟨ξ, v − v

⟩< −ρ ;

implies (1.6).

In the following remark, the special case of affine forcing terms is analyzed, providingfurther simplification.

Remark 3.6. If dK is C1 on a neighborhood of ∂K and, for a subset U ⊂ Y ,

(3.9) F (t, x) = f0(t, x) + g(t, x)U,

where Y is a Banach space,

f0 : I ×X → X and g : I ×X → L(Y,X),

then the classical inward pointing condition implies (1.5) with F and also with co Fwhenever either U is compact or Y is reflexive. Namely, assume (3.4), and (2.8), (2.9) fortime independent kR, φ ∈ R+, with F replaced by f0 and g. If either U is compact or Yis reflexive, then co F (t, x) = f0(t, x) + g(t, x)co U . Then the inward pointing condition:

∀ x ∈ ∂K, ∀ t ∈ I, ∃ u ∈ U such that 〈∇dK(x), f0(t, x) + g(t, x)u〉 < 0(3.10)

implies (1.5) both with F and with co F . Indeed, by compactness of [0, 1] and continuityof f0(·, x) and g(·, x), assumption (3.10) yields:

∀ x ∈ ∂K, ∃ ρ > 0, ∀ t ∈ I, ∃ u ∈ U satisfying 〈∇dK(x), f0(t, x) + g(t, x)u〉 < −ρ.(3.11)

Let t ∈ I and u ∈ co U be so that 〈∇dK(x), f0(t, x) + g(t, x)u〉 ≥ 0. Thus

〈∇dK(x), f0(t, x)〉 ≥ −〈∇dK(x), g(t, x)u〉.Then, taking u as in (3.11), we obtain

〈∇dK(x),f0(t, x) + g(t, x)u−(f0(t, x) + g(t, x)u

)〉

= 〈∇dK(x), g(t, x)u− g(t, x)u〉 ≤ 〈∇dK(x), g(t, x)u+ f0(t, x)〉 < −ρ,yielding (1.5) with F and also with co F .

Under the same assumptions and F given by (3.9), let us consider two examples, wherecondition (3.10) can be further simplified.Case 1: 0 ∈ U . If 〈∇dK(x), f0(t, x)〉 < 0, for any x ∈ ∂K and t ∈ I, then (3.10) holds foru = 0.Case 2: U is the unit sphere in Y . Here, if

(3.12) 〈∇dK(x), f0(t, x)〉 < ‖g(t, x)∗∇dK(x)‖Y 6= 0,

for any x ∈ ∂K and t ∈ I, then (3.10) holds for

u = − g(t, x)∗∇dK(x)

‖g(t, x)∗∇dK(x)‖Y,(3.13)

where g(t, x)∗ is the adjoint of g(t, x).


Indeed, in this case, for any x ∈ ∂K,

〈∇dK(x), f0(t, x) + g(t, x)u〉 = 〈∇dK(x), f0(t, x)〉 − ‖g(t, x)∗∇dK(x)‖Y < 0,

yielding (3.10).

4. The case of Hilbert spaces

Here we analyze the case when the state space X is Hilbert. In this setting, we show thatif the state constraint is convex then the inward pointing condition can be drasticallysimplified by involving projections on convex sets instead of generalized gradients of theoriented distance function which do belong to the dual space X∗. This turns out to bevery useful in the applications, as we will show in Section 5.

Let 〈·, ·〉X be the scalar product in X and let K be a proper closed subset of X suchthat K = IntK. Denote by Z the set of points z ∈ Xr∂K admitting a unique projectionP∂K(z) on ∂K. This set is dense in X (see [26]). For every z ∈ Z, set

nz =z − P∂K(z)

‖z − P∂K(z)‖Xsgn(dK(z)) .

A new inward pointing condition involving nz is proposed in this Hilbert framework inorder to obtain results analogous to those from Section 3.

Theorem 4.1. Assume (2.6)–(2.9). Then,

(i) the assertions of Theorem 3.1 are valid under the following inward pointing con-dition:


∀ v ∈ coF (t, x) satisfying supτ≤η, z∈Z∩B(x,η)

〈nz, S(τ) v〉X ≥ 0,

∃ v ∈ coF (t, x) ∩B(v,M) satisfying supτ≤η, z∈Z∩B(S(τ)x,η)

⟨ξ, S(τ) (v − v)

⟩X< −ρ.

(ii) the assertions of Theorem 3.2 are valid under the following inward pointing con-dition:


∀ v ∈ F (t, x) satisfying supτ≤η, z∈Z∩B(x,η)

〈nz, S(τ) v〉X ≥ 0, for some z0 ∈ B(x, η),

∃ v ∈ F (t, x) ∩B(v,M) satisfying supτ≤η, z∈Z∩B(S(τ)x,η)

⟨nz, S(τ) (v − v)

⟩X< −ρ.

Again, these conditions can be simplified when the data satisfy some compactnessassumptions.

Proposition 4.2. Assume that (2.8)–(2.9) with time independent kR, φ ∈ R+ and (3.4)hold true. Further suppose that either (3.5) is valid, or F (t, x) is convex for any t ∈ Iand x ∈ ∂K, and

∀ x ∈ ∂K, ∃ r > 0 such that the setnz : z ∈ Z ∩B(x, r)

is pre-compact.(4.3)


Then, the following assumption: for any x ∈ ∂K, there exists ρ > 0 such that

for any t ∈ I and v ∈ F (t, x) satisfying infε>0

supz∈Z∩B(x,ε)

〈nz, v〉X ≥ 0,(4.4)

there exists v ∈ F (t, x) satisfying infε>0

supz∈Z∩B(x,ε)

⟨nz, v − v

⟩X< −ρ .

implies (4.2).

Remark 4.3. The proof of Proposition 4.2 provided in Section 6 implies that it is stillvalid if (4.3) is replaced by the following less restrictive assumption:

for x ∈ ∂K define N (x) :=Limsupz→x, z∈Z nz (the Kuratowski upper limit) and as-sume that for all x ∈ ∂K the set N (x) is compact and for every ε > 0 there exists δ > 0such that

nz ∈ N (x) + εB ∀ z ∈ Z ∩B(x, δ).

In particular, if ∂K is of class C1, then the above holds true.

4.1. Convex state constraints. If K convex, then the inward pointing conditions (4.1),(4.2), and (4.4) can be weakened by replacing Z with Kc := X \K. Indeed, any z ∈ Kc

admits a unique projection on ∂K and, as proved in the following proposition, for anyz ∈ IntK ∩ Z we can find an element w ∈ Kc such that nz = nw.

Proposition 4.4. Let K be a closed convex set such that K = IntK. Then, for anyz ∈ IntK ∩ Z, there exists w ∈ Kc such that z − P∂Kz = P∂Kw − w. In particular,nz = nw.

Proof. Let z ∈ IntK∩Z and P∂K(z) be its unique projection on ∂K. By the Hahn-Banachtheorem, there exists p ∈ X such that ‖p‖X = 1 and

〈p, P∂K(z)〉X ≤ 〈p, k〉X , for any k ∈ K.

Let

M+ =x ∈ X : 〈p, x− P∂K(z)〉X ≥ 0

⊇ K

and

M = ∂M+ =x ∈ X : 〈p, x− P∂K(z)〉X = 0

.

Then M is a closed hyperplane in X and there exists a unique projection PM(z) of z onM. Actually, since P∂K(z) ∈M,

‖z − PM(z)‖X ≤ ‖z − P∂K(z)‖X ,

and, since K ⊂M+ and z lies in the interior of K,

‖z − PM(z)‖X ≥ ‖z − P∂K(z)‖X ,

we deduce that PM(z) = P∂K(z). Take w = z + 2(P∂K(z)− z). As z ∈ IntK ⊂ IntM+,we have

〈p, w − P∂K(z)〉X = 〈p, P∂K(z)− z〉X < 0,

yielding w ∈ X \M+ ⊂ Kc. Further, for any x ∈M,

0 = 〈z−PM(z), x−PM(z)〉X = 〈z−P∂K(z), x−P∂K(z)〉X = 〈P∂K(z)−w, x−P∂K(z)〉X .


This implies that P∂K(z) = PM(w). Finally, sinceM+ is a closed convex set, w admits aunique projection PM+(w) = PM(w) = P∂K(z). So, for any k ∈ K ⊂M+,

〈w − P∂K(z), k − P∂K(z)〉X ≤ 0,

implying P∂K(z) = P∂K(w). This ends the proof.

5. Examples

The examples analyzed in this section describe some classical models involving partial dif-ferential equations and integrodifferential equations, to which we may apply our abstractresults. In all the examples the state constraints satisfy the positive invariance property(2.6).

5.1. A one-dimensional heat equation. The first example is a one-dimensional par-abolic equation describing the heat flux in a cylindrical bar, whose lateral surface isperfectly insulated and whose length is much larger than its cross-section. The Neumannboundary conditions are assumed, corresponding to the requirement that the heat fluxat the two ends of the bar is zero. For x = x(t, s) : [0, 1] × [0, 1] → R we consider thefollowing inclusion (we omit the variable s in the sequel)

(5.1)

x(t) ∈ Ax(t) + F (t, x(t)), t ∈ [0, 1]x(0) = x0.

The state space is X = H1(0, 1) and the linear operator acting as Ax = x′′−x with domainD(A) =

x ∈ H2(0, 1;R) : x′(0) = x′(1) = 0

is the infinitesimal generator of a strongly

continuous semigroup S(t) : X → X, see e.g. [29, chapter II]. (The notation prime standsfor the distributional derivative.) Classical results in PDEs ensure that, if the initialdatum x0 takes nonnegative values, then the solution x to x(t) = Ax(t), x(0) = x0 takesnonnegative values. The reader is referred to [1] or [27], containing a number of examplesof sets invariant under the action of the semigroup associated with A. In particular, ifthe state constraint is the cone of nonnegative functions:

x(t) ∈ K =x ∈ X : x ≥ 0

,

then the invariance property (2.6) is satisfied. Moreover, K is convex and IntK 6= ∅. Thestate space is endowed with the scalar product

〈x, y〉X = x(0)y(0) + 〈x′, y′〉L2(0,1), for any x, y ∈ X,

whose associated norm

‖x‖2X = |x(0)|2 + ‖x′‖2

L2(0,1) for any x ∈ X,

turns out to be equivalent to the usual one ‖ · ‖H1(0,1). We show next that the set

(5.2)nz : z ∈ Kc

is pre-compact.

Since K is a closed convex cone, then any z ∈ Kc can be uniquely represented as

z = P∂K(z) + b(z),

with b(z) ∈ K−, here K− is the negative polar cone to K. By [32],

K− =p ∈ X : p′ is nondecreasing and p(0) ≤ p′(s) ≤ 0, for a.e. s ∈ [0, 1]

,


see also [24] where an explicit formula for b is provided. To prove (5.2), notice thatnz =

b(z)

‖b(z)‖X: z ∈ KC

⊂ Q :=

p

‖p‖X: p ∈ K−, p 6= 0

⊂ ∂B.

Any y ∈ Q satisfies y′ nondecreasing and

−1 ≤ y(0) ≤ y′(s) ≤ 0, for a.e. s ∈ [0, 1].

So, taking a sequence yn in Q,

‖yn‖W 1,∞(0,1) := ‖yn‖L∞(0,1) + ‖y′n‖L∞(0,1) ≤ 3,

implying that yn(0) → y(0) (up to a subsequence). Since y′n is nondecreasing, Helly’sselection theorem, see [20], allows to deduce that, (again up to a subsequence),

y′n(s)→ g(s), for a.e. s, with g ∈ L∞(0, 1),

and, applying Lebesgue dominated theorem, we deduce that yn → g in L2(0, 1). Further,

yn(s) = yn(0) +

∫ s

0

y′n(τ)dτ → y(s) := y(0) +

∫ s

0

g(τ)dτ, for any s ∈ [0, 1].

Hence g = y′, y ∈ W 1,∞(0, 1) ⊂ H1(0, 1), yielding the required pre-compactness.Let F satisfy assumptions (2.8)–(2.9) with time independent kR, φ ∈ R+, (3.4), and let

F (t, x) be convex, for any x ∈ ∂K and t ∈ I. Taking into account Proposition 4.2 and theresults in subsection 4.1, the inward pointing condition (4.2) is implied by the followingassumption: for any x ∈ ∂K there exists ρ > 0 such that

for any t ∈ I and v ∈ F (t, x) satisfying infε>0

supz∈Kc∩B(x,ε)

〈nz, v〉X ≥ 0,

there exists v ∈ F (t, x) satisfying infε>0

supz∈Kc∩B(x,ε)

⟨nz, v − v

⟩X< −ρ .

5.2. Fourier’s problem of the ring. In the second example we consider the temperaturedistribution in a homogeneous isotropic circular ring with diameter small in comparisonwith its length and perfectly insulated lateral surfaces. This problem can be modeled bya one-dimensional equation with periodic boundary conditions

(5.3)

x(t) ∈ Ax(t) + F (t, x(t)), t ∈ [0, 1]x(0) = x0,

where x = x(t, s) : [0, 1]× [0, 1]→ R (s is omitted as in the previous example), the statespace is X = H1

per(0, 1) :=x ∈ H1(0, 1;R) : x(0) = x(1)

, the linear operator acting

as Ax = x′′ with domain D(A) = H2(0, 1;R) ∩H1per(0, 1) is the infinitesimal generator of

a strongly continuous semigroup S(t) : X → X, see e.g. [9]. As before we supplementinclusion (5.3) with the state constraint

x(t) ∈ K =x ∈ X : x ≥ 0

.

Then, K satisfies condition (2.6), see for instance [23] dealing with invariant sets forsemigroups. Again, K is a closed and convex set with non empty interior. Hence, by theresults contained in subsection 4.1, the inward pointing conditions (4.1), (4.2), and (4.4)hold true with Z replaced by Kc.


5.3. A model for Boltzmann viscoelasticity. The last example deals with the phe-nomena of isothermal viscoelasticity. An integrodifferential inclusion is involved, since,as outlined in the seminal works of Boltzmann and Volterra [7, 8, 30, 31], a correct de-scription of the mechanical behavior of elastic bodies requires the notion of memory. Thekey assumption in this theory is that both the instantaneous stress and the past stressesinfluence the evolution of the displacement function y = y(x, t) : Ω × R → R. HereΩ ⊂ R3, a bounded domain with smooth boundary ∂Ω, represents the region occupied bythe elastic body. Omitting the variable x in the sequel, we study the following inclusion

(5.4) y(t) + A[y(t)−

∫ ∞0

µ(s)y(t− s) ds]∈ F(t, y(t)), t > 0,

where, A = −∆ with domain D(A) = H2(Ω) ∩H10 (Ω), according to the assumption that

the body is kept fixed at the boundary of Ω, the memory kernel µ, taking into accountthe viscoelastic behavior, is supposed to be a (nonnegative) nonincreasing and summablefunction on R+, with total mass

κ =

∫ ∞0

µ(s)ds ∈ (0, 1),

piecewise absolutely continuous, and thus differentiable almost everywhere with µ′ ≤ 0.Equation (5.4) is supplemented with the following initial condition

y(0) = y0, y(0) = z0, y(−s)|s>0 = φ0(s),

for some prescribed data y0, z0, φ0, the latter taking into account the past history of y.Applying Dafermos’ history approach, see [13], we can write (5.4) as a differential inclusionof type (1.1). To this aim, we first introduce an auxiliary variable which contains all theinformation about the unknown function up to the actual time

ηt(s) = y(t)− y(t− s), t ≥ 0, s > 0

and we recast problem (5.4) as the system of two variables y = y(t) and η = ηt(s)

(5.5)

y(t) + A[(1− κ)y(t) +

∫ ∞0

µ(s)ηt(s)ds]∈ F(t, y(t)),

ηt = Tηt + y(t).

with initial conditions

y(0) = y0, y(0) = z0, η0 = η0 = y0 − φ0.

Here the operator T is the infinitesimal generator of the right-translation semigroup onthe memory space M = L2

µ(R+;H10 (Ω)), namely,

Tη = −η′ with domain dom(T ) =η ∈M : η′ ∈M, η(0) = 0

.

The notation prime standing for the distributional derivative, and η(0) = lims→0 η(s)in H1

0 (Ω). In [11], details and related bibliography can be found, jointly with someapplications of this model to optimal control problems. Now, defining the linear operatorA on the state space X = L2(Ω)×H1

0 (Ω)×M, acting as

A(y, z, η) =

(z,−A

[(1− κ)y +

∫ ∞0

µ(s)η(s) ds], T η + z

)


with domain

dom(A) =

(y, z, η) ∈ X

∣∣∣ z ∈ H10 (Ω) , η ∈ dom(T ),

(1− κ)y +∫∞

0µ(s)η(s)ds ∈ H2(Ω) ∩H1

0 (Ω)

and setting

x(t) = (y(t), z(t), η(t)), x0 = (y0, z0, η0), F (t, x(t)) =(0,F(t, y(t)), 0

),

we view (5.5) as the following problem in X:

(5.6)

x(t) ∈ Ax(t) + F (t, x(t)), for t ∈ Ix(0) = x0.

The operator A generates a strongly continuous semigroup of contractions S(t) : X → Xwhose first component is a solution of (5.4) for F = 0. Here, taking as a state constraintin (5.6)

K = B,

we deduce that (2.6) is satisfied. Further, let U, f0 and g be as in Remark 3.6. SinceK = B in a Hilbert space, an appropriate adaptation on the basis of Section 4 of theinward pointing condition (3.10) reads: for any x ∈ X with ‖x‖ = 1 and any t ∈ I, thereexists u ∈ U such that

〈x, f0(t, x) + g(t, x)u〉 < 0.

In particular in the case when U is the unit sphere in RN , the condition

〈x, f0(t, x)〉 < ‖g(t, x)∗x‖RN 6= 0,

implies the above inward pointing condition, for u defined by u = − g(t,x)∗x‖g(t,x)∗x‖RN

. As dis-

cussed in Remark 3.6, Proposition 3.5 holds in this case implying the validity of Theorems3.1 and 3.2.

Appendix

This section contains some technical results needed in the course of the investigation. Thefirst is an infinite dimensional version of the Filippov Theorem, see [15, Theorem 1.2],modified here in a suitable form for our scopes.

Lemma A.1. Let δ0 ≥ 0 and t0 ∈ I. Assume (2.7)–(2.8), let y be a solution to (2.3) in[t0, 1], for some f ∈ L1(t0, 1;X). Set R = 1

2maxt∈[t0,1] ‖y(t)‖X ,

γ(t) = dist(f(t), F (t, y(t))

)and m(t) = MSe

MS

∫ tt0kR(s)ds

.

If m(1)(δ0 +

∫ 1

t0γ(s)ds

)< R

2, then, for any y0 ∈ y(t0) + δ0B and any β > 1, there exists

a solution x to (1.1) in [t0, 1], satisfying x(t0) = y0,

‖x(t)− y(t)‖X ≤ m(t)(δ0 + β

∫ t

t0

γ(s)ds), for any t ∈ [t0, 1],

and

‖fx(t)− f(t)‖X ≤ kR(t)m(t)(δ0 + β

∫ t

t0

γ(s)ds)

+ βγ(t), for a.e. t ∈ [t0, 1],


where fx ∈ L1(t0, 1;X) is so that (2.1) and (2.2) hold true for f = fx.

Proof. The proof proceeds exactly as in [15, Theorem 1.2]. The only difference is in thefirst line of page 109, while applying Lemma 1.3 of [15]. The assumption β > 1 is neededto ensure that, for a.e. t, the set

F (t, y(t)) ∩ f(t) + βγ(t)B 6= ∅.

Indeed, if γ(t) = 0, the definition of γ ensures that f(t) ∈ F (t, y(t)), while, if γ(t) > 0,since β > 1, from the very definition of distance and the measurable selection theorem weget that there exists a measurable selection w(t) ∈ F (t, y(t)) such that ‖f(t)−w(t)‖X ≤βγ(t). Thus

w(t) ∈ F (t, y(t)) ∩ f(t) + βγ(t)B.Notice that in finite dimension we can take β = 1, recovering the original Filippov Theo-rem.

The second result is a version of the mean value theorem for the oriented distance dKin Hilbert spaces. Here we make use of the notations introduced in Section 4.

Lemma A.2. Let (X, 〈 , 〉X) be a Hilbert space. For every x, y ∈ X we have

dK(y)− dK(x) ≤ infε>0

supz∈Z∩([x,y]+εB)

〈nz, y − x〉X .

Proof. Let x, y ∈ X and fix ε > 0. It is enough to consider the case x 6= y. For everys ∈ [0, 1] set γ(s) = x+ s(y − x). Then dK(γ(·)) is absolutely continuous. It is sufficientto prove that for almost every s ∈ [0, 1]

d

dsdK(γ(s)) ≤ sup

z∈Z∩B(γ(s),ε)

〈nz, y − x〉X .

Indeed, then it follows that

dK(y)− dK(x) =

∫ 1

0

d

dsdK(γ(s)) ds ≤ sup

z∈Z∩([x,y]+εB)

〈nz, y − x〉X .

The maps dK(γ(·)) and d2K(γ(·)) are almost everywhere differentiable. Denote by D ⊂

(0, 1) the set on which both functions are differentiable and fix s ∈ D. We distinguishthree cases.

Case 1: γ(s) /∈ K. Let L > 0 be a Lipschitz constant for d2K on γ([0, 1]) + B. Recall

that |dK(x)| = dist(x, ∂K), for all x ∈ X. Then for each 0 < h < minε, 1− s and everyz ∈ Z ∩ (B(γ(s), h2) rK) we have

(A.1) d2K(γ(s)) ≥ ‖z − P∂K(z)‖2

X − Lh2

and

d2K(γ(s+ h)) ≤ ‖γ(s+ h)− P∂K(z)‖2

X = ‖γ(s)− z‖2X

+ 2〈γ(s)− z , z − P∂K(z) + h(y − x)〉X + ‖z − P∂K(z) + h(y − x)‖2X

= ‖z − P∂K(z) + h(y − x)‖2X + o(h) .


Consequently, for all h > 0 small enough,

d2K(γ(s+h))−d2K(γ(s))

h≤ infz∈Z∩(B(γ(s),h2)rK)

‖z−P∂K(z)+h(y−x)‖2X−‖z−P∂K(z)‖2Xh

+ o(1)

= infz∈Z∩(B(γ(s),h2)rK) 2 〈z − P∂K(z) , y − x〉X + o(1)

= infz∈Z∩(B(γ(s),h2)rK) 2 dK(γ(s)) 〈nz , y − x〉X + o(1)

≤ supz∈Z∩B(γ(s),ε) 2 dK(γ(s)) 〈nz , y − x〉X + o(1) .

Then we obtain

d

dsd2K(γ(s)) ≤ sup

z∈Z∩B(γ(s),ε)

2 dK(γ(s)) 〈nz , y − x〉X

andd

dsdK(γ(s)) =

1

2 dK(γ(s))· d


z∈Z∩B(γ(s),ε)

〈nz , y − x〉X .

Case 2: γ(s) ∈ Int K. Similarly to Case 1, for every 0 < h < minε, s and everyz ∈ Z ∩ (B(γ(s), h2) rKc) we have (A.1) and

d2K(γ(s− h)) ≤ ‖γ(s− h)− P∂K(z)‖2

X

(A.2)

= ‖γ(s)− z‖2X + 2〈γ(s)− z , z − P∂K(z)− h(y − x)〉X + ‖z − P∂K(z)− h(y − x)‖2

X

= ‖z − P∂K(z)− h(y − x)‖2X + o(h) .

Inequalities (A.1) and (A.2) yield

d2K(γ(s− h))− d2

K(γ(s))

h≤ inf

z∈Z∩(B(γ(s),h2)rKc)2 〈P∂K(z)− z , y − x〉X + o(1)

= infz∈Z∩(B(γ(s),h2)rKc)

2 (−dK(γ(s))) 〈nz , y − x〉X + o(1)

≤ 2 (−dK(γ(s))) supz∈Z∩B(γ(s),ε)

〈nz , y − x〉X + o(1) .

Then we obtain

d

dsd2K(γ(s)) ≥ 2 dK(γ(s)) sup

z∈Z∩B(γ(s),ε)

〈nz , y − x〉X

and again

d

dsdK(γ(s)) =

1

2 dK(γ(s))· d


z∈Z∩B(γ(s),ε)

〈nz , y − x〉X .

Case 3: γ(s) ∈ ∂K. Let us first suppose that | ddsdK(γ(s))| = 2C > 0. Then for all

h > 0 small enough we have |dK(γ(s + h))| ≥ Ch, so that the point s is isolated in theset s ∈ D : γ(s) ∈ ∂K. Consequently,

µ

(s ∈ D : γ(s) ∈ ∂K ,

d

dsdK(γ(s)) 6= 0

)= 0 .


It remains to verify that in the case where ddsdK(γ(s)) = 0, we have

supz∈Z∩B(γ(s),ε)

〈nz , y − x〉X ≥ 0 .

Assume by contradiction that there exists α > 0 such that

supz∈Z∩B(γ(s),ε)

〈nz , y − x〉X ≤ −α .

Let 0 < h < ε/4‖y − x‖X , ξii∈N ⊂ B(γ(s), ε2) ∩ Int K converge to γ(s) as i→ +∞ and

set, for every 0 ≤ r ≤ h, γi(r) = ξi + r(y − x). Fix i ∈ N and define

hi = sup0 < h < h : γi([0, h]) ⊂ Int KProceeding as in Case 2, we can prove that

d

drdK(γi(r)) ≤ sup

z∈Z∩B(γi(r),ε4

)

〈nz , y − x〉X ≤ −α for a.e. r ∈ [0, hi] .

Then, γi(hi) ∈ Int K and, consequently, hi = h. Summarizing, we have obtained that forevery i ∈ N

d

drdK(γi(r)) ≤ sup

z∈Z∩B(γi(r),ε4

)

〈nz , y − x〉X ≤ −α for a.e. r ∈ [0, h] .

Therefore, for every i ∈ N and h ∈ (0, ε4‖y−x‖X

) we have dK(γi(h)) − dK(γi(0)) ≤ −αh .Taking the limit as i→ +∞, we obtain

dK(γ(s+ h))− dK(γ(s)) ≤ −αh .This is impossible, since d

dsdK(γ(s)) = 0.

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CNRS, Institut de Mathematiques de Jussieu - Paris Rive Gauche, UMR 7586,Sorbonne Universites, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cite,Case 247, 4 Place Jussieu, 75252 Paris, FranceE-mail address: [email protected]

Dipartimento di Matematica “F.Brioschi”, Politecnico di MilanoPiazza Leonardo da Vinci 32, 20133 Milano, ItalyE-mail address: [email protected]

Sorbonne Universites, UPMC Univ Paris 06, Institut de Mathematiques de Jussieu -Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cite,Case 247, 4 Place Jussieu, 75252 Paris, FranceE-mail address: [email protected]

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